reserve S for non empty satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct;
reserve a,b for POINT of S;
reserve A for Subset of S;
reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct;
reserve a,b,c,m,r,s for POINT of S;
reserve A for Subset of S;

theorem Th8:
  for a,b,c,d,e being POINT of S st b <> c & c <> d & between b,c,d &
  (between a,b,c or between b,a,c) & (between c,d,e or between c,e,d) holds
  between a,c,e
  proof
    let a,b,c,d,e be POINT of S;
    assume that
A1: b <> c and
A2: c <> d and
A3: between b,c,d and
A4: between a,b,c or between b,a,c and
A5: between c,d,e or between c,e,d;
    between a,c,d by A1,A3,A4,GTARSKI3:18,19;
    hence thesis by A2,A5,GTARSKI3:17,19;
  end;
